1.2.1 Naming and Measuring Angles
- 1. Naming and Measuring Angles The student will be able to (I can): • Correctly name an angle • Classify angles as acute, right, or obtuse • Use the Angle Addition Postulate to solve problems
- 2. angleangleangleangle – a figure formed by two rays or sides with a common endpoint. Example: vertexvertexvertexvertex – the common endpoint of two rays or sides (plural: vertices). Example: A is the vertex of the above angle ● ● ● A C R
- 3. Notation: An angle is named one of three different ways: 1. By the vertex and a point on each ray (vertex must be in the middle) : ∠TEA or ∠AET 2. By its vertex (if only one angle): ∠E 3. By a number: ∠1 Methods 1 and 3 are always correct. Method 2 can only be used if there is only one angle at that vertex. ● ● ● E T A 1
- 4. Example Which name is notnotnotnot correct for the angle below? ∠TRS ∠SRT ∠RST ∠2 ∠R ● ● ● S R T 2
- 5. Example Which name is notnotnotnot correct for the angle below? ∠TRS ∠SRT ∠RST ∠2 ∠R ● ● ● S R T 2
- 6. acuteacuteacuteacute angleangleangleangle – an angle whose measure is greater than 0° and less than 90°. rightrightrightright angleangleangleangle – an angle whose measure is exactly 90°. obtuseobtuseobtuseobtuse angleangleangleangle – an angle whose measure is greater than 90° and less than 180°.
- 7. straightstraightstraightstraight angleangleangleangle – an angle whose measure is exactly 180° (also known as opposite rays, or a line) ●
- 8. congruentcongruentcongruentcongruent anglesanglesanglesangles – angles that have the same measure. m∠WIN = m∠LHS ∠WIN ≅ ∠LHS Notation: “Arc marks” indicate congruent angles. Notation: To write the measure of an angle, put a lowercase “m” in front of the angle bracket. ●● ● ● ● ● L H S W IN m∠WIN is read “the measure of angle WIN”
- 9. interior of aninterior of aninterior of aninterior of an angleangleangleangle – the set of all points between the sides of an angle Angle AdditionAngle AdditionAngle AdditionAngle Addition PostulatePostulatePostulatePostulate: If D is in the interiorinteriorinteriorinterior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC (part + part = whole) Example: If m∠ABD=50˚ and m∠ABC=110˚, then m∠DBC=60˚ ● ●● ● A B D C
- 10. Example The m∠PAH = 125˚. Solve for x. m∠PAT + m∠TAH = m∠PAH 2x + 8 + 3x + 7 = 125 5x + 15 = 125 5x = 110 x = 22 ● ●● ● P A T H (3x+7)˚ (2x+8)˚
- 11. angleangleangleangle bisectorbisectorbisectorbisector – a ray that divides an angle into two congruent angles. Example: UY bisects ∠SUN; thus ∠SUY ≅ ∠YUN or m∠SUY = m∠YUN ● ●● ● S U N Y
- 12. Examples ∠PUN is bisected by UT, m∠PUT = (3+5x)° and m∠TUN = (3x+25)°. What is m∠PUN? m∠PUT = m∠TUN 3 + 5x = 3x +25 2x = 22 x = 11 m∠PUN = 2(3 + 5(11)) = 116° ● ● ● ● P U N T
- 13. Example Point R is in the interior of ∠NFL. If m∠NFR = (7x – 1)° and m∠RFL = (3x+23)°, what value of x would make FR an angle bisector? If FR is going to be an angle bisector, then m∠NFR = m∠RFL 7x – 1 = 3x + 23 4x = 24 x = 6 Therefore, if x = 6, then FR is an angle bisector.